an:01113820
Zbl 0898.11045
Adleman, Leonard M.; Huang, Ming-Deh A.
Counting rational points on curves and abelian varieties over finite fields
EN
Cohen, Henri (ed.), Algorithmic number theory. Second international symposium, ANTS-II, Talence, France, May 18-23, 1996. Proceedings. Berlin: Springer. Lect. Notes Comput. Sci. 1122, 1-16 (1996).
1996
a
11Y16 11G25 11G20 14K15 14G05 14G15 14Q05 14Q15 14H25
abelian variety over a finite field; deterministic algorithm; characteristic polynomial of the Frobenius endomorphism; Jacobian varieties; rational points; hyperelliptic curves; primality test; complexity analysis
Let \(A\) be an abelian variety of dimension \(g\) over a finite field \({\mathbb F}_q\). Suppose that \(A\) is given as a closed subvariety of projective \(n\)-space. The authors exhibit a deterministic algorithm that computes the characteristic polynomial of the Frobenius endomorphism of \(A\) that runs in time \(O((\text{log} q)^c)\), where \(c\) is a polynomial expression in \(g\) as well as \(n\). This improves upon an earlier result of \textit{J. Pila} [Math. Comput. 55, 745-763 (1990; Zbl 0724.11070)], who obtained a similar result but with the constant \(c\) depending exponentially on \(n\).
By applying this to the Jacobian varieties of curves \(X\) over \({\mathbb F}_q\), one obtains a deterministic algorithm to count the number of \({\mathbb F}_q\)-rational points of \(X\) that runs in time \(O((\text{log} q)^c)\), where \(c\) is a polynomial expression in \(n\), as well as the genus \(g\) of \(X\). In the special case of hyperelliptic curves of genus \(g\), the authors show that the number of \({\mathbb F}_q\)-rational points on \(X\) can be counted deterministically in time \((\text{log} q)^{O(g^6)}\). This case is of interest in view of the primality test decribed by the authors in their monograph [Primality testing and abelian varieties over finite fields, Lect. Notes Math. 1512 (Springer-Verlag, 1992; Zbl 0744.11065)].
For the entire collection see [Zbl 0852.00023].
Ren?? Schoof (Amsterdam)
Zbl 0744.11065; Zbl 0724.11070